| Abstract: | This paper considers filtered polynomial approximations on the unit sphere (mathbb {S}^dsubset mathbb {R}^{d+1}), obtained by truncating smoothly the Fourier series of an integrable function f with the help of a “filter” h, which is a real-valued continuous function on ([0,infty )) such that (h(t)=1) for (tin [0,1]) and (h(t)=0) for (tge 2). The resulting “filtered polynomial approximation” (a spherical polynomial of degree (2L-1)) is then made fully discrete by approximating the inner product integrals by an N-point cubature rule of suitably high polynomial degree of precision, giving an approximation called “filtered hyperinterpolation”. In this paper we require that the filter h and all its derivatives up to (lfloor tfrac{d-1}{2}rfloor ) are absolutely continuous, while its right and left derivatives of order (lfloor tfrac{d+1}{2}rfloor ) exist everywhere and are of bounded variation. Under this assumption we show that for a function f in the Sobolev space (W^s_p(mathbb {S}^d), 1le ple infty ), both approximations are of the optimal order ( L^{-s}), in the first case for (s>0) and in the second fully discrete case for (s>d/p), conditions which in both cases cannot be weakened. |
